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Preprint submitted on 26 Nov 2020
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Exact splitting methods for semigroups generated by
inhomogeneous quadratic differential operators
Joackim Bernier
To cite this version:
Joackim Bernier. Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators. 2020. �hal-02425591v2�
EXACT SPLITTING METHODS FOR SEMIGROUPS GENERATED BY INHOMOGENEOUS QUADRATIC DIFFERENTIAL OPERATORS
JOACKIM BERNIER
Abstract. We introduce some general tools to design exact splitting methods to com-pute numerically semigroups generated by inhomogeneous quadratic differential opera-tors. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker-Planck equations.
Contents
1. Introduction 1
2. Exact splitting methods 6
3. Exact classical-quantum correspondance 11
4. Applications 17
5. Appendix 25
References 32
1. Introduction
1.1. Motivation. Before presenting the general setting of this paper, let us motivate our splitting methods on a quick example. Assume that we aim at solving numerically the following evolution equation on Rn
∂tu(t, x) = −(|x|2− ∆) u(t, x), t ≥ 0, x ∈ Rn
u(0, x) = u0(x), x∈ Rn,
where n ≥ 1 and u0 is a smooth and well localized function on Rn. Since the Hermite
functions diagonalize the harmonic oscillator, a first natural approach, to reach spectral accuracy, consists in using a Fast Hermite Transform. However, if the solution has to be known in the space variables (for example on a grid) at each time step, such a method is quite costly.
A cheaper classical method consists in splitting the harmonic oscillator. For example, here, a natural splitting would be the following Strang splitting
(1) e−δt(|x|2−∆)≃ e−δt|x|2/2eδt∆e−δt|x|2/2.
At each time step, this method requires only a Fast Fourier Transform to solve the heat equation. Unfortunately, even if it reaches spectral accuracy in space, it is only second
order in time. Furthermore, note that, here, the only way to get classical splitting methods of higher order is to use complex time steps (see e.g. [11,12]) and that the higher the order of the method is, the larger the number of FFTs per time-step is (and so the more costly the method is).
However, allowing the sub-time-steps of the splitting method to be nonlinear functions of δt, the following splitting formula can be established (see subsection 7.4 of [2] for a
proof)
(2) e−δt(|x|2−∆)= e−12tanh(δt)|x|2e12sinh(2δt)∆e−12tanh(δt)|x|2.
Note that, contrary to the Strang splitting (1), this factorization is exact : there is no remainder term. Consequently, it is much more accurate than (1) and the time step can be taken quite large (the only possible restrictions coming from the spatial discretization). Furthermore, it is as cheap as (1) to compute.
Roughly speaking, in this paper, we explain how and why the evolution equations asso-ciated with a large class of operators called inhomogeneous quadratic differential operators, can be solved by some exact splitting methods. These exact splittings are similar to (2): they provide numerical methods of infinite order (spectral in space and exact in time) using only a small number of FFTs. Sometimes the coefficients of these splittings are not given by an explicit formula depending on δt (like in (2)), nevertheless, in any case, we provide
an iterative method to compute them efficiently.
We warn the reader that the example of the exact splitting (2) may be somehow mislead-ing about our methods. They are not just a generalization of the classical splittmislead-ing where the sub-time-steps would become nonlinear functions of δt(as in (2)). Most of the time, the
exact splitting requires the computation of solutions of evolution equations we would not compute in a classical splitting method. For example, to solve the Fokker–Planck equation
∂tu(t, x, v) + v∂xu(t, x, v) = ∂v(v + ∂v)u(t, x, v), x, v∈ R, t ≥ 0,
u(0, x, v) = u0(x, v), x, v∈ R,
with an exact splitting, we have to solve some Schrödinger equations (see Proposition 4.7
for details).
1.2. General context and result. We consider the problem of the numerical resolution by splitting methods of linear partial differential equations of the form
(3)
∂tu(t, x) = −pwu(t, x), t≥ 0, x ∈ Rn
u(0, x) = u0(x), x∈ Rn
where n ≥ 1, u0 ∈ L2(Rn) and pw is an inhomogeneous quadratic differential operator
acting on L2(Rn).
First, let us precise what we hear by inhomogeneous quadratic differential operator. This terminology is used (for example by Hörmander in [20]) to denote the Weyl quantization of a polynomial function of degree 2 or less. More precisely, if p is a polynomial function on C2n of degree 2 or less (called symbol) whose decomposition in coordinates is given by
p(X) =tXQX +tY X + c
where X = t(x1, . . . , xn, ξ1, . . . , ξn), Q is a symmetric matrix of size 2n with complex
operator acting on L2(Rn) defined by pw= t x −i∇ Q x −i∇ +tY x −i∇ + c.
Note that, for such polynomials, this formula coincides with the usual (and much more general) formula defining the Weyl quantization through oscillatory integrals (see e.g., Section 18.5 in [21] or Chapter 1 in [23]).
The Weyl quantization allows in many situations the deduction of some properties of the operator from properties of its symbol. For example, as stated in the following proposition, if the real part of p is bounded below on R2n then (3) is globally well posed.
Proposition 1.1. If p is a polynomial of degree 2 or less on C2nwhose real part is bounded below on R2n then −pw generates a strongly continuous semigroup on L2(Rn) denoted by
(e−tpw)t∈R+.
This proposition is very classical and relies on the Hille-Yosida Theorem. For example, a proof is given by Hörmander in [20] (pages 425-426) when p is quadratic, but its proof can clearly be extended to the case where p is inhomogeneous (see e.g. Theorem 4.7 in [20]).
We recognize that the class of equations given by (3), where p is a polynomial of degree 2 or less, may seem too elementary to require the use of specific methods to solve them. However, it is usual to have to solve them as sub-steps of splitting methods for more sophisticated equations. Thus it is crucial to have robust methods to compute them very efficiently. Furthermore, for many of these models, in some relevant regimes, their dynamics are the leading part of the dynamics. So, it is crucial to be able to compute them with as much accuracy as possible.
For example, the linear part of some nonlinear Schrödinger equations describing some rotating Bose–Einstein condensates (see e.g. [5,8]) is of the form (3) where
pw= (−i|ξ|2− iV (x) − iBx · ξ)w= i∆− Bx · ∇ − iV (x).
where V : Rn→ R is a quadratic external potential and B is a real skew symmetric matrix of size n associated with a constant external magnetic field. The formalism of (3) allows also to consider transport equations associated with affine vectors fields. Indeed, if B is a square real matrix of size n and y∈ Rn then
(iBx· ξ + iy · ξ + TrB)w = (Bx + y)· ∇.
Even if such transport equations are essentially trivial, their resolution is required to com-pute, using splitting methods, the solutions of many kinetic equations (e.g. the Vlasov-Maxwell equations with a constant external magnetic field, see [9,13]). Finally, equations like (3) can also describe some phenomena of diffusions. For example, the generalized Ornstein–Uhlenbeck operators are of the form
(ξ· Aξ + iBx · ξ + x · Rx + TrB)w =−∇ · A∇ + Bx · ∇ + x · Rx
where A, R are some real nonnegative symmetric matrices of size n and B is a real matrix n. Note that these operators include Fokker–Planck and Kramers–Fokker–Planck operators (see e.g. [16,19]).
Some of the equations of the form (3) are easy to solve numerically using pseudo-spectral methods. For example, to solve the heat equation or to compute a shear, it is enough to do some Fast Fourier Transforms. Similarly, in the spirit of the splitting methods, it is not very costly to solve successively some of these equations. More precisely, let us define, in this context, the operators we consider as easily computable using standard pseudo-spectral methods.
Definition 1.2. An operator acting on L2(Rn) can be computed by an exact splitting if it can be factorized as a product of operators of the form
eα∂xj, eiαxj, eia(∇), eia(x), eαxk∂xj, e−b(x), eb(∇), eγ
where α ∈ R, γ ∈ C, a, b : Rn → R are some real quadratic forms, b is nonnegative,
j, k ∈ J1, nK and k 6= j. As usual, a(∇) (resp. b(∇)) denotes the Fourier multiplier associated with −a(ξ) (resp. −b(ξ)), i.e. a(∇) = (−a(ξ))w.
Note that if an operator can be computed by an exact splitting then it is bounded. The following theorem justifies why we focus on splitting methods for semigroups generated by inhomogeneous quadratic differential operators.
Theorem 1.3. If p is a polynomial of degree 2 or less on C2n whose real part is bounded
below on R2n then e−pw can be computed by an exact splitting.
Remark 1.4. Actually, we prove a slightly stronger result : the quadratic forms associated with a and b in the definition of Definition 1.2can be chosen diagonal. Nevertheless, it is not necessarily relevant from a numerical point of view because the sub-steps required to diagonalize the quadratic forms can be costly.
This theorem is proved in the subsection 5.2of the Appendix. Unfortunately, its proof does not provide an efficient way to split semigroups (minimizing, for example, the number of sub-steps or the number Fast Fourier Transforms required to approximate the exponen-tials). Nevertheless, as illustrated in Section4, on many examples, paying attention to the particular structure of each semigroup, we are able to design optimized exact splittings.
This work can be considered as the theoretical part of a more general study. Indeed, a second work [10], written in collaboration with Nicolas Crouseilles and Yingzhe Li, deals with the implementation of these methods and compares numerically their efficiency, their accuracy and their qualitative properties with respect to the existing methods. We also couple these exact splitting methods with classical methods in order to solve some nonlinear equations and some non quadratic linear equations.
Outline of the work . In Section 2, we develop the notion of exact splitting in a more general framework and specify its links with the classical splittings. It will naturally lead to a general theorem to design efficient exact splittings for many linear ordinary differential equations. In Section 3, we explain how the theory of the Fourier Integral Operators developed by Hörmander in [20] can be used to transform exact splittings of linear ordinary differential equations into exact splittings of semigroups. And, finally, in Section 4, we apply the results of the previous sections to obtain some efficient exact splittings for the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker–Planck equations.
Acknowledgment . The author thanks P. Alphonse, N. Crouseilles and Y. Li for many en-thusiastic discussions about this work. Research of the author was supported by ANR project NABUCO, ANR-17-CE40-0025.
Notations and conventions. Let us define some classical notations used in this paper. • To get convenient notations, most of the time, we denote by t (instead of δt) the
time-step of our methods.
• In denotes the identity matrix on Rn and J2n denotes the matrix of the canonical
symplectic form of R2n, i.e. J2n:= In −In .
• By convention, the empty spaces in the matrix notations refer to coefficients equal to zero (see for example the definition of J2n above).
• If A is a matrix tA denotes the transpose of A.
• If K ∈ {R, C}, Mn(K) denotes the algebra of the square matrices of size n and
Sn(K) = {M ∈ Mn(K), tM = M} denotes the vector space of the symmetric
matrices.
• For K ∈ {R, C}, we will use the following classical groups of matrices
GLn(K) ={M ∈ Mn(K), M is invertible} SLn(K) ={M ∈ Mn(K), det M = 1}
Sp2n(K) ={M ∈ M2n(K), tM J2nM = J2n} On(R) ={M ∈ Mn(R), tM M = In}.
and their associated Lie algebras
gln(K) = Mn(K) sln(K) ={M ∈ Mn(K), TrM = 0}
sp2n(K) ={J2nQ, Q∈ S2n(K)} son(R) ={M ∈ Mn(R), tM =−M}.
where Tr denotes the trace and the Lie bracket is formally defined through the relation [A, B] := AB− BA.
• A real Lie algebra of matrices is a sub-Lie-algebra of gln(R) for some n≥ 0.
• We equip the space of the polynomials of degree 2 or less on C2n of its structure of Lie algebra induced by the canonical Poisson bracket. This one being defined for two polynomials p1, p2 on C2n ≡ Cnx× Cnξ by {p1, p2} := n X j=1 ∂ξjp1∂xjp2− ∂xjp1∂ξjp2.
Note that if K ∈ {R, C}, the space of the quadratic forms on K2n is a Lie algebra
naturally isomorphic to sp2n(K).
• If g is a Lie algebra, ad : g → g denotes its adjoint representation, i.e. ∀x, y ∈ g, adxy := [x, y].
• We consider the natural action of the analytic functions on Mn(C) defined by
the holomorphic functional calculus (see VII-3 of [17] for details). By abuses of notations, if f is analytic on a domain Ω ⊂ C and M ∈ Mn(C) has its spectrum
included in Ω then (f (z))(M ) is just an other way to denote f (M ). Note that, if f : D(zc, ρ)→ C is an analytic function on the complex disk of center zcand radius
ρ > 0 and M ∈ Mn(C) has its spectrum included in D(zc, ρ) then f (M ) can be
defined by the convergent series f (M ) =X
k∈N
f(k)(zc)
k! (M − zcIn)
k.
• S(Rn) denotes the Schwartz space on Rn.
• In a non-commutative setting, the notation Q to denote a product can be quite ambiguous. In this paper, we adopt the following natural convention. If I is a totally ordered finite set and (gj)j∈I ∈ MI, where M is a monoid1, then
Y
j∈I
gj := gι1. . . gι♯I
where ι is the increasing bijection from J1, ♯IK to I. 2. Exact splitting methods
The original setting of the splitting methods (see e.g [18]) consists in considering linear equations2 of the form
(4)
∂tu = Lu = L(1)u +· · · + L(k)u
u(t = 0) = u0.
whose solution is denoted u(t) = exp(tL)u0 and such that
exp(tL(1)), . . . , exp(tL(k)) are nicer than exp(tL).
In this context, nicer usually means cheaper to compute. Then the approximations of u at times tn= nδtare got compositing n times an approximation Ψδt of exp(δtL) where Ψδt is a composition of operators of the form exp(ασjδtL
(σj)) with 1≤ σ
j ≤ k and αj ∈ R. The
most classical methods are the Lie splitting where Ψ(Lie)δ
t = exp(δtL
(1)) . . . exp(δ tL(k))
and the Strang splitting Ψ(Strang)δ t = exp( δt 2L (1)) . . . exp(δt 2L (k−1)) exp(δ tL(k)) exp( δt 2L (k−1)) . . . exp(δt 2L (1)).
These methods are respectively of order 1 and 2. It means that they provide respectively approximations of order δt and δt2 of u(tn). Similar methods can be derived to obtain
splitting methods of arbitrarily high order. However, note that, the higher the order is, the higher the number of step is and so the higher the cost of the method is. Furthermore, the only way to get methods of order higher than 2 where the ασj are nonnegative is to allow the ασj to be complex. In this case it is possible to design schemes of high order where the real parts of the ασj are nonnegative (see [11, 12]). Note that this is crucial when irreversible equations are considered.
The setting of the exact splitting is the same as the setting of the classical ones. Nev-ertheless, we have to assume that
1i.e. a set equipped with an associative binary operation and an identity element.
2Note that as usual this formalism include nonlinear equations, since it is enough to consider the
exp(tL(1)), . . . , exp(tL(k)) are nicer than exp(tL) because L(1), . . . , L(k) belong to some vector spaces E1, . . . , Ek.
This assumption is natural because usually the exp(tL(j)) are nice due to a particular
structure of each L(j) (e.g. nilpotent, diagonal...). Consequently, the idea of the exact splitting consists in looking for Ψδt as a product of operators of the form exp(δtL
(σj)
j,δt ) with 1 ≤ σj ≤ k, L(σj,δjt) ∈ Eσj and to ask that Ψδt is exactly equal to exp(δtL). In other words, we are looking for a factorization of exp(δtL) as a product of operators of the form
exp(δtL(σj,δjt)) with Lj,δ(σjt)∈ Eσj.
Remark 2.1. To avoid any possible confusion note that exp(δtL(σj,δjt)) does not denote the
solution of a non-autonomous equation but the solution of ∂tu = L(σj,δjt)u at time t = δt. Actually, the existence of an exact splitting can be seen as an inverse problem with respect to the classical backward error analysis of splitting methods. In the context of the splitting method, this analysis is realized through the Baker-Campbell-Hausdorff formula (see [18]). It states that
(5) exp(ασ1δtL
(σ1)) . . . exp(α
σℓδtL
(σℓ)) = exp(δ
tΩδt,L,α,σ)
where Ωδt,L,α,σcan be expanded in powers of δtand the operator associated with the power δn
t is obtained as a linear combination of n Lie brackets of the L(j). For example, we have
Ωδt,L,α,σ= ℓ X j=1 ασjL (σj)+δt 2 X i<j ασjασi[L (σi), L(σj)] +O(δ2 t)
where [L(σi), L(σj)] = L(σi)L(σj)− L(σj)L(σi). Note that, in general, the formula (5) and the expansion of Ωδt,L,α,σ have to be understood in the sense of the formal series in δt. Nevertheless, if L(1), . . . L(k)belong to a real Lie algebra of matrices then these expansions converge when δt is small enough (see e.g. [18]).
Now if we consider a product of operators of the form exp(δtL(σj,δjt)) with 1 ≤ σj ≤ k, L(σj)
j,δt ∈ Eσj, where the Ej are some vector subspaces of a same real Lie algebra of matrices then the Baker-Campbell-Hausdorff formula states that it is of the form
(6) exp(ασ1δtL
(σ1)
1,δt) . . . exp(ασℓδtL
(σℓ)
ℓ,δt) = exp(δtΩeδt,L,σ),
where eΩδt,L,σ admits an expansion similar to the expansion of Ωδt,L,α,σ. Consequently, to get an exact splitting method we just have to design L(σℓ)
j,δt in order to cancel all the Poisson brackets in the expansion of eΩδt,L,σ, i.e. we have to solve
(7) Ωeδt,L,σ =
k
X
j=1
L(j).
Since it can be shown that eΩδt,L,σ is smooth with respect to δt and (L
(σj)
j,δt)1≤j≤ℓ, it is natural to try to solve (7) with the Implicit Function Theorem.
In this way, we prove the following theorem for which we have many applications. From now, to get convenient notations, we denote t instead of δt.
Theorem 2.2. Let m be a positive integer and b1, . . . , bm, s be some subspaces of a real
Lie algebra of matrices such that b1, . . . , bm are complementary. If b⋆ = b⋆,1+· · · + b⋆,m∈
b1⊕ · · · ⊕ bm= b is such that
(8) (b1⊕ · · · ⊕ bm) + adb⋆(s) is a real Lie algebra
then there exist t0 > 0 and an analytic function t ∈ (−t0, t0) 7→ (bt, st) ∈ b × s such that
b0 = b⋆ and
(9) ∀t ∈ (−t0, t0), etb⋆ = e−tstetbt,1. . . etbt,metst.
Remark 2.3. Before proving this theorem, let us do some remarks.
• In most of the applications, s can be chosen as a one of the bj. Consequently, assuming
that s = b1, the notations of this theorem are consistent with the notations previously
introduced in this section through the identifications δt = t, k = m, Ej = bj, b⋆ =
L(1)+· · · + L(m), ℓ = k + 2, L(1) 1,δt =−st, L (j−1) j,δt = bt,j−1, L (1) ℓ,δt = stwhere j∈ J2, m + 1K. • Since the proof of this theorem relies on the Implicit Function Theorem, the coefficients of the exact splitting (i.e. stand bt) can be efficiently computed by an iterative method.
Unfortunately, this method require some notations introduced in the proof of Theorem
2.2to be presented. Consequently, it is introduced just after the proof.
• In practice, as we will see in Section4, it may be useful to use Theorem2.2just to deter-mine a priori the form of an exact splitting and then to get analytically the associated coefficients (using for example a formal computation software).
• The assumption (8) of Theorem 2.2is a bit too strong. The optimal assumption seems that eΩδt,L,σ (defined implicitly by (6)) belongs to the vector space defined in (8) for all L and δt. Paying attention to the expansion of eΩδt,L,σ given by the Baker-Campbell-Hausdorff formula, it essentially means that we do not need to ask this space to contain Lie brackets of elements belonging to a same space bj. Note that, such a generalization
would be necessary to justify a priori the form of the exact splitting of the Example 2.3 of [2] for the Kramer–Fokker–Planck operator.
Proof of Theorem 2.2. Naturally, following the assumption of the theorem, we introduce the real Lie algebra of matrices, denoted g and defined by
(10) g= (b1⊕ · · · ⊕ bm) + adb⋆(s).
Applying the Baker Campbell Hausdorff formula, we get a neighborhood of the origin in b× s, denoted V , and an analytic function F : V → g such that for all real t and all (b, s)∈ b × s such that t(b, s) ∈ V , we have
e−tsetb1. . . etbmets = exp tb + t2[b, s] + t2 2 X 1≤i<j≤m [bi, bj] + t3F (tb, ts) . Consequently, we aim at solving the equation
(11) b⋆ = b + tadb(s) + t 2 X 1≤i<j≤m [bi, bj] + t2F (tb, ts).
To solve this equation we have to introduce some notations. First, from the decompo-sition (10) of g, we deduce naturally that there exists a complementary space to b in g denoted r, a subspace of s denoted s′ and a vector space isomorphism Ψ : s′→ r such that
(12) Ψ = Πr◦ adb⋆
where Πb, Πr are the canonical projections associated with the decomposition
(13) g= b⊕ r.
Then, let s⋆ ∈ s′ be defined by
(14) s⋆ =− 1 2(Ψ −1◦ Π r) X 1≤i<j≤m [b⋆,i, b⋆,j] , let V⋆ be a neighborhood of (b⋆, s⋆) in b× s′ and let t⋆> 0 be such that
(−t⋆, t⋆)V⋆ ⊂ V.
So, to solve (11), we are going to apply the Implicit Function Theorem in (0, b⋆, s⋆) to the
function G : (−t⋆, t⋆)× V⋆ → b× s′ (t, b, s) 7→ (Gb, Gs′)(t, b, s) where Gb(t, b, s) = b− b⋆+ t ΠbR(t, b, s). with R(t, b, s) = [b, s] + 1 2 X 1≤i<j≤m [bi, bj] + tF (tb, ts), and ΨGs′(t, s(m), b, s(r)) = Πr◦ adbs +1 2Πr X 1≤i<j≤m [bi, bj] + t ΠrF (tb, ts).
First, observe that, by construction of G, if G(t, b, s) = 0 then b, s is a solution of (11). Indeed, by construction, the equation (11) can be rewritten as
(Gb+ tΨGs′)(t, b, s) = 0. Then observe that, by construction of s⋆ we have
G(0, b⋆, s⋆) = 0.
Consequently, since G is clearly an analytic function, to conclude the proof applying the Implicit Function Theorem, we just have to prove that the partial differential of G with respect to (b, s) in (0, b⋆, s⋆) is invertible. Indeed, using a natural matrix representation,
this differential is
Ib
Lb,⋆ Is′
where Lb,⋆ : b → s′ is an explicit linear map depending on b⋆ and s′. This matrix being
triangular, it is clearly invertible and its invert is
Ib
−Lb,⋆ Is′
Now, let us present an iterative method to determine the coefficients bt and stgiven by
the Theorem 2.2.
Proposition 2.4. Assume that the assumption of Theorem 2.2is satisfied, let Ψ be defined by (12), s⋆ be defined by (14) and Πb, Πr be the projections canonically associated with the
decomposition (13).
There exists t1 ∈ (0, t0), such that for all t ∈ (−t1, t1) the sequence (b(k)t , s (k)
t )k∈N if well
defined by induction as follow :
Initially, we have b(0)t = b⋆ and s(0)t = s⋆ and for k≥ 0
(15) ( b(k+1)t = b(k)t + b⋆ − Πbg(k) s(k+1)t = s(k)t − t−1Ψ−1Π rg(k) where g(k)= t−1log(e−ts(k)t etb (k) t,1 . . . etb (k) t,mets (k) t ).
Furthermore, there exist C > 0 independent of t and k such that for all k ≥ 0 and all t∈ (−t1, t1), we have
|b(k)t − bt| + |s(k)t − st| ≤ C 2−k.
Note that, here, as usual, the log function denotes the principal determination of the matrix logarithm. We have chosen to present an iterative method as elementary as possible. However, computing the Lie brackets associated with the operators Ls,⋆ and Lb,⋆ it would
be possible to get a natural iterative method whose convergence rate would be τk instead of 2−k with τ a linear function of|t|.
Proof of Proposition 2.4. It follows from the proof of the theorem2.2that the sequence is defined by a relation of the kind
(b(k+1)t , s(k+1)t ) = Wt(b(k)t , s (k) t )
where for all t ∈ (−t0, t0), Wt is a smooth function on a same ball B of center (b⋆, s⋆)
and t 7→ Wt is smooth. Consequently, to prove that the sequence is well defined if |t| is
small enough and that Wt has a fix point we just have to prove that Wt is a contraction
mapping for a well chosen norm. Indeed, it follows of the proof of the theorem 2.2 that the differential of W0 in (b⋆, s⋆), denoted dW0(b⋆, s⋆), is
0b
−Lb,⋆ 0s′
.
Thus, since it is nilpotent, applying for example the Lemma 5.6.10 of [22], we get a matrix norm k · k⋆ such that
kdW0(b⋆, s⋆)k⋆ ≤ 1/4.
Now, since W is smooth on (−t0, t0)× B and W0(b⋆, s⋆) = (b⋆, s⋆), we deduce of the mean
value inequality that there exist t2 ∈ (0, t0) and a ball B⋆, for the norm k · k⋆, centered
Consequently, Wt has an unique fix point inB⋆ and the sequence (b(k)t , s (k)
t ) converges to
this fix point with the rate 2−k.
Finally, we conclude this proof observing that, by construction and continuity, there exists t1 ∈ (0, t2) such that for all t∈ (−t1, t1), (bt, st) belongs to B⋆ and is a fix point of
Wt.
3. Exact classical-quantum correspondance
As we have seen in Section2, exact splittings can be designed (using for example Theo-rem2.2) to solve linear ordinary differential equations. We aim at designing exact splittings to solve some partial differential equations. So, first, it is natural to focus on linear trans-port equations. Indeed, the formula of the characteristics
(16) et Bx·∇u = u◦ etB
where u∈ L2(Rn) and B is square matrix of size n, provides a natural way to transform
an exact splitting at the level of the linear ordinary differential equations into an exact splitting at the level of the linear transport equations.
For example, we have the following exact splitting for the two-dimensional rotations
(17) exp 0 tan(θ/2) 0 0 exp 0 0 − sin θ 0 exp 0 tan(θ/2) 0 0 = exp(θJ2),
where θ∈ (−π, π). At the level of the associated transport equation, this formula can be written
et(x2∂x1−x1∂x2)= etan(t/2)x2∂x1e− sin(t)x1∂x2etan(t/2)x2∂x1.
This factorization is very useful to compute rotations since, using semi-Lagrangian meth-ods, it only requires one dimensional interpolations instead of a two dimensional interpo-lations as we could expect. The formula is well known in image processing and has been used for decades to rotate images (see e.g. [24]). Two recent papers deal with the appli-cations of this decomposition for the numerical resolution of kinetic equations (see [3,9]). This factorization has also been extended to compute 3 dimensional rotations with only one dimensional interpolations (see e.g. [14,29]). Note that, in Subsection 4.1, we extend this kind of decomposition in any dimension and for more general transforms (including rotations and dilatations).
The inhomogeneous quadratic differential operators and the semigroups they generate enjoy some strong and specific properties providing a more general way to transform an exact splitting at the level of the linear ordinary differential equations into an exact splitting at the semigroup level.
Indeed, usually, the Lie bracket of two differential operators is also a pseudo-differential operator. Its symbol is given by the Moyal bracket of the symbols. In many applications, the Moyal bracket admits a natural expansion whose leading part3 is given
by the Poisson bracket of the symbols (see e.g. [21]). Furthermore, in the particular case of the inhomogeneous quadratic operators all the higher order terms vanish and we have (18) [q1w, q2w] =−i{q1, q2}w
where q1, q2 are some polynomials of degree 2 or less on C2n. 3in some specific sense depending on the problem.
The formula (18) is especially useful to realize some changes of unknowns in order to put some operators in normal form (see e.g. [7] for an application to the reducibility that can also be seen as a factorization of exponentials). Note that, applying formally the Baker–Campbell–Hausdorff formula at the level of the operator, (18) suggests a way to transform an exact splitting at the level of the linear ordinary differential equations into an exact splitting at the level of the semigroups generated by inhomogeneous quadratic forms. We refer the reader to the beginning of the Section 3 of [2] for details about this heuristic. It is used in [15, 6, 5] to design some exact splitting methods to solve linear Schrödinger equations with harmonic potentials and rotating terms. The result suggested by these formal computations is made rigorous in Proposition3.6 below, and relies on the representation of the semigroups as Fourier Integral Operators (detailed in Theorem 3.5
below).
To present this notion, we need to introduce some basic notations and associated prop-erties.
Definition 3.1. T is a non-negative complex symplectic linear bijection on C2n, and we denote T ∈ Sp+2n(C), if T ∈ Sp2n(C) and
∀X ∈ C2n, tXtT (−iJ2n)T X−tX(−iJ2n)X∈ R+.
Note that this set is naturally equipped with a structure of monoid (i.e. it is stable by multiplication and the identity belongs to Sp+2n(C)).
Definition 3.2. If q is a quadratic form on C2n then its Hamiltonian flow at time t∈ R, denoted by Φqt, is the flow at time t of the linear ordinary differential equation
(19) d
dtz =−iJ2n∇q(z) where ∇q =t(∂x1q, . . . , ∂xnq, ∂ξ1q, . . . , ∂ξnq).
Note that, in particular, the linear ordinary differential equation (19) can be solved using an exponential and thus we have
(20) Φqt = e−2itJ2nQ
where Q is the matrix of q. The following proposition summarizes some elementary prop-erties of these Hamiltonian flows that will be used all along this paper.
Proposition 3.3. Let q1, q2, q be some quadratic forms on C2n and T ∈ Sp2n(C) then the
following properties holds i) ∀t ∈ R, Φqt ∈ Sp2n(C),
ii) {q1, q2} = 0 ⇐⇒ ∀t ∈ R, Φqt1Φtq2 = Φqt1+q2,
iii) ∀t ∈ R, T−1Φq
tT = Φq◦Tt .
Usually, in this context, the second property is called Noether’s theorem because it is also equivalent to have q1◦ Φqt2 = q1 for all t∈ R. Proofs of these properties can be found,
for example, in a nonlinear context, in [18].
The following classical lemma (whose proof is recalled in the subsection5.3 of the Ap-pendix) links naturally the two previous definitions.
Lemma 3.4. If q is a quadratic form on C2n such that ℜq is nonnegative on R2n then
∀t ≥ 0, Φqt ∈ Sp+2n(C).
The following theorem, proved by Hörmander in [20] (Thm 5.12 and Prop 5.9), is the main tool we use to realize exact splittings.
Theorem 3.5 (Hörmander [20]). There exists a map K : Sp+
2n(C)→ B/U2
where U2 = {+1, −1} and B is the unit ball of L (L2(Rn)), the algebra of the bounded
operators on L2(Rn), which is monoid morphism, i.e.
∀S, T ∈ Sp+2n(C), K (ST ) = K (S)K (T ) and K(I2n) =±idL2.
Furthermore, if q is a quadratic form on C2n such that ℜq is nonnegative on R2n then
K(Φq
t) =±e−tq
w .
In [20], K is defined through an explicit but heavy formula that is not relevant for us here. An operator of the form K (T ) with T ∈ Sp+2n(C) is called a Fourier Integral
Operators. It provides a natural way to transform an exact splitting at the level of the linear ordinary differential equations into an exact splitting at the level of the semigroups generated by quadratic differential operators (up to an argument of continuity to remove the uncertainty of sign).
To the best of our knowledge, the idea to use the Fourier integral operators to get an exact splitting has been introduced by Paul Alphonse and the author in [2]. We aimed at characterizing the regularizing effects of the semigroups generated by non-selfadjoint quadratic differential operator. Our splitting provides a quite explicit representation of the polar decomposition of this semigroup. Note that it is the decomposition we would get applying Theorem 2.2and Theorem 3.5with s ={0}, b1= i sp2n(R) and b2 = sp2n(R).
In order to give a corollary of Theorem 3.5 well suited to get exact splittings for semi-groups generated by inhomogeneous quadratic differential operators, we have to introduce a last elementary notation. If p : C2n = Cn
x× Cnξ → C is a polynomial of degree 2 or less
that we write in coordinates as
p = tXQX +tY X + c
where X = t(x1, . . . , xn, ξ1, . . . , ξn), Q ∈ S2n(C), Y ∈ C2n and c ∈ C, then Pp : C2n+2 =
Cn+1x × Cn+1
ξ → C is the complex quadratic form defined by
Pp = tXQX +tY Xxn+1+ cx2n+1.
Proposition 3.6. Let p1,t, . . . , pm+1,t : C2n → C be some polynomials of degree 2 or less
depending continuously on t∈ [0, t0] for some t0> 0, whose real part is uniformly bounded
below on R2n and satisfying
(21) ∀t ∈ [0, t0], ΦPpt 1,t. . . ΦPp m,t t = ΦPpt m+1,t then we have (22) ∀t ∈ [0, t0], e−tp w 1,t. . . e−tpwm,t = e−tpwm+1,t.
The proof is given at the end of this section. Let us mention that this proposition is also a corollary of the Theorem 2.3 of [28]. Note that if the polynomials are homogeneous (i.e. if they are some quadratic forms) then the P can be removed. Furthermore, as a corollary of the proof, if we do not assume that the polynomials are uniformly bounded below and that the polynomials depend continuously on t, then (22) holds up to an uncertainty of sign.
It may also be interesting to have exact splittings to compute evolution operators gen-erated by non-autonomous inhomogeneous quadratic differential operators (like it is done for the non-autonomous linear magnetic Schrödinger equations in [5]). Here, we chose for conciseness to do not consider the non-autonomous case. Nevertheless, let us mention that it would be possible to generalize Proposition 3.6to deal with non-autonomous equations using the generalization of the Theorem3.5proven in [25]. With such a generalization, the exponential of matrices become naturally the solutions of non-autonomous linear ordinary differential equations (which can be studied similarly using Magnus expansions, see [18]).
In order to illustrate Proposition 3.6, let us give an elementary application (Section
4 being devoted to more sophisticated applications). Indeed, the formula (17) used to compute rotations can clearly be extended analytically for θ ∈ −2it where t ∈ R. Since, up to a transposition, this formula can be written as
Φt(tanh t)/(2t) x2Φ(sinh 2t)/(2t) ξt 2Φt(tanh t)/(2t) x2 = Φ|x|t 2+|ξ|2
we deduce of the Proposition 3.6 that the exact splitting (2) for the harmonic oscillator (presented in the introduction) holds.
Before focusing on the proof of Proposition3.6, let us present the following Lemma that is useful in practice to establish the factorization (21). It expresses the triangular nature of the equation (21). Its proof is postponed in subsection5.1of the appendix.
Lemma 3.7. Let p1, . . . , pm be some polynomials of degree 2 or less on C2n whose
decom-positions are
pj = qj+ ℓj + cj
where qj quadratic forms, ℓj are linear forms and cj are complex numbers. Then we have
(23) ΦPp1 1 . . . ΦPp1 m = ΦPp1 m+1 if and only if (24) Φq1 1 . . . Φ qm 1 = Φ qm+1 1 m X j=1 ℓj◦ Υqj Y k>j Φqk 1 = ℓm+1◦ Υqm+1 m X j=1 cj+ κj+ σj = cm+1+ κm+1
where, denoting Q(j) the matrix of qj and L(j)∈ C2n the matrix of ℓj
Υqj = ez− 1 z (−2iJ2nQ(j)), κj = X k∈N 4k (2k + 3)! qj((J2nQ (j))kJ 2ntL(j)),
σj =− i 2 j−1 X p=1 L(p)Υqp j−1Y k=p+1 Φqk 1 ΥqjJ2n tL(j).
In this lemma, it is relevant to note that if q1, . . . , qm+1 satisfy the first equation of (24)
then the second equation is just a linear system with respect to ℓ1, . . . , ℓm+1and that if the
two first equations are satisfied then the last equation is linear with respect to c1, . . . , cm+1.
In order to prove Proposition 3.6, we introduce some technical lemmas that will be crucial.
Lemma 3.8. If p : R2n → R is a real polynomial of degree two or less being bounded below then there exists a nonnegative real quadratic form q : R2n→ R
+, Y ∈ R2n and c∈ R such
that p can be written as
(25) p(X) = q(X− Y ) + c
where X = (x1, . . . , xn, ξ1, . . . , ξn).
Proof of Lemma 3.8. Naturally p can be written in coordinates as p(X) = tXQX +tZX + p(0)
where Q∈ S2n(R) is the matrix of q and Z ∈ R2n. Now realizing a canonical factorization,
we observe that it is equivalent to prove that p can be written as (25) and that Z∈ Im Q. In other words, we just have to prove that
∀X0 ∈ Im Q⊥, tX0Z = 0.
But since if X0∈ Im Q⊥ we have
p(λX0) = λtX0Z + p(0)
and we know, by assumption, that λ 7→ p(λX0) is bounded below then we deduce that tX
0Z = 0, which conclude this proof.
Corollary 3.9. If p : R2n → R is a real polynomial of degree two or less being bounded below then we have
P(p− inf
R2np)≥ 0
Proof of Corollary 3.9. Applying Lemma3.8, p can be written as p(X) = q(X− Y ) + c.
So, by definition, we have
P(p− c) = q(X − xn+1Y ).
Thus, since q is nonnegative, P(p− c) is also nonnegative and c is the infimum of p. Lemma 3.10. If ψ ∈ S(Rn+1) and p is a polynomial of degree 2 or less on C2n whose real
part is nonnegative on R2n then ℜPp is nonnegative on R2n+2, e−(Pp)wψ ∈ S(Rn+1) and we have
(e−(Pp)wψ)|Rn×{1}= e−p w
Proof of Lemma 3.10. First, observe that Corollary3.9proves directly thatℜPp is nonneg-ative. Consequently, the semigroup generated by −(Pp)w is well defined (see e.g.
Propo-sition1.1).
Applying the Theorem 4.2 of [20], we know that
∀t ∈ [0, 1], φ(t) = e−t(Pp)wψ∈ S(Rn+1) and that φ∈ C∞(R+
t × Rn+1) and satisfies
∀t ∈ R+, ∂tφ =−(Pp)wφ.
Now decomposing p by homogeneity as
p = q + ℓ + c
where q : R2n → C is a complex valued quadratic form, ℓ : R2n → C is a complex valued
linear form and c∈ C, we have
∀t ∈ R+, ∂tφ =−(qw+ xn+1ℓw+ cx2n+1)φ.
Since here we deal with smooth functions, this relation can be evaluated in xn+1= 1. Thus
we get
∀t ∈ R+, ∂tρ =−(qw+ ℓw+ c)ρ =−pwρ
where ρ = φ|R+
t×Rn×{1}. Consequently by definition of the semigroup exp(−tp
w), we have
ρ = exp(−tpw)ρ(0,·) = exp(−tpw)ψ|Rn×{1}.
Finally, evaluating this last relation for t = 1, we get the desired relation. Using these two lemmas, we give the following proof of the Proposition3.6.
Proof of Proposition 3.6. In order to obtain a factorization of the semigroup using Theorem
3.5, we have to deal with quadratic forms having a nonnegative real part. So, since the polynomials are uniformly bounded below, we apply Corollary 3.9to get a constant c > 0 such that
(26) ∀j ∈ J1, m + 1K, ∀t ∈ [0, t0], ℜ P(pj,t+ c)≥ 0.
Since the real part of P(pj,t+ c) = Ppj,t+ cx2n+1 is nonnegative, we know by lemma 3.4
that
∀j ∈ J1, m + 1K, ∀t ∈ [0, t0], ΦPp
j,t+cx2n+1
t ∈ Sp+2n+2(C).
Then observing that, by construction Ppj,t does not depend on ξn+1, it commutes with
x2n+1, i.e.
{Ppj,t, x2n+1} = 0.
where {·, ·} stands for the usual Poisson bracket associated with the canonical symplectic form on Rn+1x × Rn+1ξ . Consequently, applying the Noether theorem4, their Hamiltonian
flows commute, i.e. for all j ∈ J1, mK and all t ∈ [0, t0] we have
ΦPpj,t+cx2n+1 t = Φ Ppj,t t Φ cx2 n+1 t and Φ Ppm+1,t+cmx2n+1 t = Φ Ppm+1,t t Φ cmx2 n+1 t , and we have ∀t ∈ [0, t0], ΦPp1,t +cx2 n+1 t . . . Φ Ppm,t+cx2n+1 t = Φ Ppt+mcx2n+1 t 4note that here it could be proven more elementarily, applying Lemma3.7.
Now, we can apply the Theorem 3.5 to deduce that for all t ∈ [0, t0], there exists εt∈ {+1, −1} such that ∀t ∈ [0, t0], e−t(Pp1,t) w−tcx2 n+1. . . e−t(Ppm,t)w−tcx2n+1= ε te−t(Ppm+1,t) w−tcmx2 n+1. To get the desired factorization, we have to check that εt = 1 and to prove that this
relation can be evaluated at xn+1= 1.
First, we focus on the sign and we observe that if t = 0 all the exponentials are equal to the identity so we have ε0 = 1. Thus, since [0, t0] is connected, we just have to prove
that t 7→ εt is continuous to deduce that εt= 1 for all t∈ [0, t0]. Let φ∈ S(Rn)\ {0} be
a Schwarz function on Rn non identically equals to zero (for example a gaussian). Since
e−t(Ppt)w−tcmx2n+1 is injective (see Theorem 2.1 and Corollary 7.9 of [2]), we now that ∀t ∈ [0, t0], ke−t(Ppm+1,t) w−tcmx2 n+1φk L2 > 0. So we have εt=h m Y j=1 e−t(Ppj,t)w−tcx2n+1φ, e−t(Ppm+1,t)w−tcmx2n+1φi L2ke−t(Ppm+1,t) w−tcmx2 n+1φk−2 L2.
But it follows for the Theorem 4.2 of [20] that t7→ e−t(Pp1,t)w−tcxn+12 . . . e−t(Ppm,t)w−tcx2n+1φ and t7→ e−t(Ppm+1,t)w−tcmx2n+1φ are continuous from [0, t
0] toS(Rn). Consequently, t7→ εt
is continuous as product of two continuous functions and we have εt≡ 1, i.e.
(27) ∀t ∈ [0, t0], e−t(Pp1,t)
w−tcx2
n+1. . . e−t(Ppm,t)w−tcx2n+1 = e−t(Ppm+1,t)w−tcmx2n+1. To conclude, we just have to prove that this relation can be evaluated in xn+1= 1.
Let φ∈ S(Rn) and choose ψ∈ S(Rn+1) such that ψ|Rn×{1}= φ.
Applying m + 1 times Lemma 3.10to (27), we deduce that ∀t ∈ [0, t0], e−t(p1,t)
w−tc
. . . e−t(pm,t)w−tcφ = e−tpwm+1,t−tcmφ. Thus, since it is clear that for all t∈ [0, t0] and all j∈ J1, mK we have
e−t(pj,t)w−tc = e−t(pj,t)we−tc and e−t(pm+1,t)w−tmc = e−t(pm+1,t)we−tmc we deduce that
∀t ∈ [0, t0], e−t(p1,t)
w
. . . e−t(pm,t)wφ = e−tpwm+1,tφ.
Finally, since the semigroups are continuous on L2(Rn) and that the previous relation holds for all φ∈ S(Rn) that is dense in L2(Rn), we get the desired factorization.
4. Applications
In this section, we apply the results of the previous sections in order to get some effi-cient exact splitting methods for the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker–Planck equations. Another paper [10], written in collaboration with Nicolas Crouseilles and Yingzhe Li, deals with the im-plementation of these methods and compares numerically their efficiency, their accuracy and their qualitative properties with respect to the existing methods.
4.1. Transport equations. As we have seen through the formula (17) two dimensional rotations can be computed efficiently as products of shear transforms.
In fact, this kind of factorization is much more general since as a classical application of the Gaussian elimination algorithm, we know that each matrix of determinant 1 is a product of shear matrices 5 (see e.g. Lemma 8.7 of [27]). Identifying as usual matrices
with linear maps on Rn, this factorization is written as follows.
Proposition 4.1. For all G∈ SLn(R), there exists m ≥ 0, α ∈ Rm, k, ℓ∈ J1, nKm such
that for all j ∈ J1, mK, kj 6= ℓj and
∀u ∈ L2(Rn), u◦ G = exp(α1xk1∂xℓ1)· · · exp(αmxkm∂xℓm)u
Note that in the context of the exact splitting, it is more natural to apply this result for G = exp(tB) where B ∈ sln(R) is a matrix whose trace vanishes.
If we focus on transforms associated with matrices of determinant different than 1, we have to deal with one dimensional dilatations. Indeed, it is clear that each matrix G ∈ GLn(R) can be factorized as a product of a matrix in SLn(R) and the diagonal
matrix diag(1, . . . , 1, det G). The following proposition provides a way to deal with positive dilatations with pseudo-spectral methods.
Proposition 4.2. For all λ > 0, we have
∀u ∈ L2(R), u(λ ·) = λ−1/2e−iελαλ∂2xe−iβλx2eiελβλ∂x2eiαλx2u, where αλ = 12 p |λ−1− 1|λ−1, β λ = 12 p |1 − λ|, ελ = 1 if λ≤ 1 and ελ=−1 else.
Proof of Proposition 4.2. We observe that if t = log λ then u(λ·) = etx∂xu = et(ixξ−12)wu.
Consequently, this proposition is a consequence of the proposition 3.6 and an elemen-tary formal computation which can be checked, for example, with a formal computation
software.
We do not provide similar formulas when λ < 0 since we do not know if it may be useful for applications. Nevertheless, since we have (see e.g. Thm 2.2.3 of [23])
(28) ∀u ∈ L2(R), u(−x) = −ieiπ2(x2−∂2x)u(x), we note that, as a consequence of Theorem 1.3, such formulas exist.
The factorization provided by the Gaussian elimination algorithm in Proposition 4.1is not, in general, the most efficient possible. The following proposition, that is an application of Theorem2.2, provides some efficient exact splittings generalizing the exact splittings for rotations (17).
Proposition 4.3. Let M be a real square matrix of size n≥ 1 such that (29)
∀i, Mi,i = 0
∃i, ∀j 6= i, Mi,j 6= 0
then there exist t0 > 0 and an analytic function (y(ℓ), (y(k))k6=j, y(r)) : (−t0, t0)→ Rn×(n+1) satisfying (30) ( y(ℓ)i = y(r)i = 0 ∀k 6= i, yk(k)= 0
such that for all t∈ (−t0, t0) we have
etM x·∇= et(y(ℓ)(t)·x)∂xi Y k6=i et(y(k)(t)·x)∂xk et(y(r)(t)·x)∂ xi.
For example, this proposition justifies a priori the form of the exact splitting used in [14,29] to compute three dimensional rotations.
Proof of Proposition 4.3. Let B :=tM be the transpose of M . We are going to prove that there exists an analytic function (y(ℓ), (y(k))k6=i, y(r)) : (−t0, t0)→ Rn×(n+1) satisfying (30)
such that for all t∈ (−t0, t0),
(31) etB = (In+ t y(ℓ)(t)⊗ ej) Y k6=j (In+ t y(k)(t)⊗ ek) (In+ t y(r)(t)⊗ ej) = ety(ℓ)(t)⊗ej Y k6=i ety(k)(t)⊗ek ety(r)(t)⊗e j.
Consequently, Proposition4.3 will be proven since it is enough to apply the characteristic formula (16) to the transpose of (31). Note that the second equality in (31) comes form the fact that the matrices are nilpotent of order 1.
In order, to prove this factorization applying the Theorem 2.2, we realize the change of unknown y(i) = y(r)+ y(ℓ) and thus the first factor of (31) becomes
ety(ℓ)(t)⊗ej = e−ty(r)(t)⊗ejety(i)(t)⊗ej.
Up to an irrelevant permutation of indices, without loss of generality we assume that i = 1. In order to apply the Theorem 2.2, we define
bk={y ⊗ ek | y ∈ Rn satisfies yk= 0}, s = b1 and b = n
M
k=1
bk
and b⋆,k = Bek⊗ ek (i.e. b⋆ = B). Consequently, to prove this proposition applying the
Theorem2.2, we just have to prove that its assumption (8) is satisfied. In our context, we are going to prove that
(32) b+ adB(s) = sln(R).
So, from now, we just focus on proving (32). But, since the inclusion ”⊂ ” is obvious, we just have to prove that
(33)
b∩ adB(s) ={0},
Let Ψ : s→ Rnbe the linear map defined by
∀s ∈ s, Ψ(s) = diag ◦ adB(s)
where diag : gln(R) → Rn is the natural map extracting the diagonal coefficients of a
matrix.
Since b = Ker diag is the space of the matrices with diagonal coefficients are equal to zero, the first relation of (33) is equivalent to have Ker Ψ = {0}. Furthermore, if Ker Ψ ={0}, then adB is injective on s and we have
dim adB(s) = dim s = n− 1.
Since, for all k, dim bk= n− 1 and (n + 1)(n − 1) = n2− 1 = dim sln(R), we deduce that
if Ker Ψ ={0} then the second relation of (33) also holds.
Finally, we just have to verify that Ker Ψ = {0}. Let y ⊗ e1 ∈ Ker Ψ where y ∈ Rn
satisfies y1 = 0. By assumption, if j6= 1 then
0 = tej[B, y⊗ e1]ej =−tejyte1Bej =−yjBj,1.
Since, by assumption, Bj,16= 0 for j 6= 1, we deduce that y = 0, i.e. Ker Ψ = {0}.
4.2. Schrödinger equations. Let v : Rn → R be a real quadratic form and B ∈ so n(R)
be a skew symmetric matrix. We aim at solving the following linear Schrödinger equation on Rn
(34) i∂tu(t, x) +
1
2∆u(t, x)− v(x)u(t, x) + iBx · ∇u(t, x) = 0. In this context, a diagonal quadratic form is defined as follow.
Definition 4.4. A quadratic form is diagonal on Rn if its matrix in the canonical basis is diagonal.
The following theorem provides an optimized splitting method to solve (34). Its proof is given at the end of this subsection.
Theorem 4.5. There exists some quadratic forms vt(r), aton Rn, a strictly upper triangular
matrix Ut∈ Mn(R), a strictly lower triangular matrix Lt∈ Mn(R) and a diagonal quadratic
form vt(ℓ) on Rn, all depending analytically on t∈ (−t0, t0) for some t0 > 0, such that for
all t∈ (−t0, t0) we have eit(∆/2−v(x))−tBx·∇ = e−itvt(ℓ)(x) n−1Y j=1 e−t(Utx)j∂xj eitat(∇) n Y j=2 e−t(Ltx)j∂xj e−itvt(r)(x)
where at(∇) denotes the Fourier multiplier of symbol −at(ξ) and (Utx)j (resp. (Ltx)j) the
jst coordinate of Utx (resp. Ltx).
The efficiency of this method is optimal, more precisely it is as cheap as a basic Lie splitting. Indeed, considering that the computational cost of this kind of method is pro-portional to the number of one dimensional Fast Fourier Transform required to implement it, the method of Theorem 4.5requires 2n 1d-FFTs6 which is the same as the elementary
6An inverse Fourier transform has the same cost as a direct one. For example, to compute the solution
of the semigroup generated by the harmonic oscillator with the factorization (2), we need 2 1d-FFTs, one to go on the Fourier side and one other to come back.
Lie splitting eit(∆/2−v(x))−tBx·∇ = t→0e −itv(x) n Y j=1 e it 2∂xj2 −(Bx)j·∂xj +O(t2).
Note that this method is more general and more efficient than the other existing exact splittings for (34). For example, the splitting (2) of Bader in [5], designed to solve (34) in dimension n = 2, requires 6 1d-FFTS.
It seems to the author that there does not exist any simple explicit formula to compute the coefficients of the exact splitting given in Theorem4.5. However, due to some particular properties of nilpotency, a simple and efficient iterative method is available to compute them. Indeed, identifying a quadratic form with its symmetric matrix in order we define, if t is small enough, we define, by induction, the following sequences
At,k+1 = At,k+ In/2− eAt,k Lt,k+1 = Lt,k+ L− eLt,k Ut,k+1 = Ut,k+ U− eUt,k Vt,k+1(m) = Vt,k(m)+ V − eVt,k(m)+2t[Dt,k, B] +t 2 2D2t,k where (At,0, Lt,0+ Ut,0, Vt,0(m)) = (In/2, B, V ), L + U = B and 2 eVt,k(m) tLet,k+ t e Ut,k+ tDt,k e Lt,k+ eUt,k+ tDt,k 2 eAt,k ! =−t−1J log(Pt,k) and Pt,k = n−1Y j=1 In+ tUt,k(j) In− t t Ut,k(j) ! In 2tAt,k In n Y j=2 In+ tL(j)t,k In− t t L(j)t,k ! × In −2tVt,k(m) In !
with L(j)t,k = (ej ⊗ ej)Lt,k, Ut,k(j)= (ej⊗ ej)Ut,k and (e1, . . . , en) the canonical basis of Rn.
Theorem 4.6. There exists τ0 ∈ (0, t0) such that if 0 < |t| < τ0 then the preceding
sequences are well defined and
|At− eAt,k| + |Lt− eLt,k| + |Ut− eUt,k| + |Vt(ℓ)+ 1 2Dt,k| + |V (r) t − V (m) t,k − 1 2Dt,k| ≤ t τ0 k . We could apply directly Theorem2.2to prove Theorem4.5. Indeed, by Proposition3.6, it is clearly enough to prove that there exist some quadratic forms vt(r), at on Rn, a strictly
upper triangular matrix Ut ∈ Mn(R), a strictly lower triangular matrix Lt ∈ Mn(R) and
t0 > 0, such that for all t∈ (−t0, t0) (35) Φi( |ξ|2 2 +v(x)+Bx·ξ) t = Φ ivt(ℓ)(x) t n−1Y j=1 Φi(Utx)jξj t Φiat(ξ) t n Y j=2 Φi(Ltx)jξj t Φiv(m)t (x) t Φ −iv(ℓ)t (x) t
where v(r)t := v(m)t − v(ℓ)t . Furthermore note that all the factors of (35) are some real
symplectic transforms belonging to Sp2n(R) and that all the right hand side flows are ex-ponentials of nilpotent matrices (it explains why we don’t have to compute exponential of matrices in the iterative method). Nevertheless, due to some convenient algebraic cance-lations, the iterative method described below is not exactly the one associated with the proof of Theorem 2.2. Consequently, here it is easier to prove Theorem 4.5 and Theorem
4.6simultaneously. However, the proofs being very similar to the proof of Theorem2.2, so it is done more informally.
Before giving the proof, let us just mention shortly the algebraic decomposition we would use in order to satisfy the assumption (8) of Theorem 2.2 to prove (35). Recalling that sp2n(R) is isomorphic to the Lie algebra of the real quadratic forms (equipped with the canonical Poisson bracket), we would simply have to observe that all real quadratic form q : R2n= Rnx× Rn
ξ → R written
q(x, ξ) = a(q)(ξ) + M(q)x· ξ + b(q)(x) where a(q), b(q) are real quadratic forms on Rn and M(q)∈ M
n(R), can be decomposed as q(x, ξ) = n−1 X j=1 (Ut(q)x)jξj + a(q)(ξ) + n X j=2 (L(q)t x)jξj + b(q)(x)−[D, B]x· x 2 + ( |ξ|2 2 + Bx· ξ + v(x), txD(q)x 2 ) . where M(q)= L(q)+ D(q)+ U(q) is the natural decomposition of M(q).
Proof of Theorem 4.5and Theorem 4.6. Similarly to the proof of Theorem 2.2, we prove that the exact splitting can be obtained from the resolution of a nonlinear equation via the Implicit Function Theorem. Consequently, Theorem 4.6is nothing but the convergence of the natural iterative method associated with the Implicit Function Theorem.
In order to obtain (35), we consider (35) as a nonlinear equation where vt(m), at, Ut, Lt, vt(ℓ)
are the unknowns. Applying the BCH formula, we get a family of real quadratic forms qt
on R2n (depending analytically on v(m)
t , at, Ut, Lt) such that if t is small enough then
n−1Y j=1 Φi(Utx)jξj t Φiat(ξ) t n Y j=2 Φi(Ltx)jξj t Φiv(m)t (x) t = Φiqtt.
Thus, since Φ−iv (ℓ) t (x)
t is symplectic, applying Proposition 3.3, we get
Φiqt◦Φ −iv(ℓ)t (x) t t = Φ i(|ξ|22 +v(x)+Bx·ξ) t .
The exponential map begin injective near the origin, we deduce that, (35) is equivalent to (36) qt◦ Φ−iv (ℓ) t (x) t = |ξ| 2 2 + v(x) + Bx· ξ. However Φ−iv (ℓ) t (x) t is a shear transform Φ−iv (ℓ) t (x) t (x, ξ) = (x, ξ + 2tV (ℓ) t ).
Consequently, decomposing naturally qt, with notations similar to the ones of the iterative
method, as
qt(x, ξ) = evt(m)(x) + (eLtx + eUtx + tDt)x· ξ + eat(ξ)
we deduce that (36) is equivalent to the system of equations
eat(ξ) =|ξ|2/2, Let= L, Uet= U, Vt(ℓ)=−Dt/2,
e
v(m)t (x) = v(x)− (eLtx + eUtx + tDt)x· 2tVt(ℓ)x− 2t2|V (ℓ) t x|2.
Noting that by substitution, the last equation writes e
vt(m)(x) = v(x) + t
2[B, Dt]x· x + t2
2Dtx· Dtx,
we conclude that to get the factorization (35), it is enough to choose Vt(ℓ)=−Dt/2 and to
solve the nonlinear equation
F (t, at, Lt, Ut, vt(m)) : = (eat(ξ), eLt, eUt, evt(m)(x)− t 2[B, Dt]x· x − t2 2Dtx· Dt) = (|ξ|2/2, L, U, v(x)).
As in the proof of Theorem 2.2, this nonlinear equation can be solved by the Implicit Function Theorem. Indeed, F (0,|ξ|2/2, L, U, v(x)) = (|ξ|2/2, L, U, v(x)), F is an analytic
function of t, at, Lt, Ut, v(m)t , and the partial differential of F in (0,|ξ|2/2, L, U, v(x)) with
respect to (at, Lt, Ut, vt(m)) is the identity. Note that the natural iterative method associated
with the resolution of this nonlinear equation with the implicit function theorem is exactly the one introduced above.
4.3. Fokker–Planck equations. We apply our exact splitting methods to two Fokker– Planck equations. These equations can be used to describe particles system with collisions (in plasma physics or astrophysics). We refer to [19, 16, 2] for more details about these equations.
4.3.1. Exact splitting for the Fokker–Planck equation. First, we focus on the classical in-homogeneous Fokker–Planck equation
(FP) ∂tu(t, x, v) + v∂xu(t, x, v) = ∂v(v + ∂v)u(t, x, v).
Note that, since this equation comes from kinetic models, the variable are not denoted x1, x2 but x, v. Implicitly, in this paper, the Fourier variable canonically associated with
Proposition 4.7. The following factorization provides an exact splitting forFP
∀t ≥ 0, e−t(v∂x−∂v2−∂vv) = et/2e−(et−1)v∂xe∇·At∇eiαt∂v2e−iβtv2e−iβt∂2veiαtv2. where αt = 12 p (1− e−t)e−t, β t = 12 √ et− 1, ∇ = t(∂
x, ∂v) and At is the positive matrix
defined by At= 1 2 e2t+ 2t + 3− 4et −4 sinh2(t/2) −4 sinh2(t/2) 1− e−2t .
A priori, it could seem strange to have to solve some Schrödinger equations in order to compute the solution of FP. However a part of the Fokker–Planck operator is associated with a dilatation, i.e. v∂v, and as explained in the previous subsection (see Proposition4.2),
the semigroup it generates cannot be computed only by compositions of shear transforms. Proof. First, let us check that At is nonnegative. Since the second diagonal coefficient of
At is positive, to check that At is nonnegative, it is enough to prove that det At ≥ 0 for
t > 0. By a some elementary formal computations, we get det At=
(3e4t− 12e3t+ (8t + 2)e2t+ 20et− 8t − 13)e−2t
16 .
Consequently, we just have to prove that (16e2tdet A
t)≥ 0. However, realizing the Taylor
expansion of 16e2tdet At in 0 (with a formal computation software), we get
16e2tdet At = t→0O(t
4).
Thus, it is enough to prove that
∂t2(e−2t∂t2(16e2tdet At))≥ 0,
which is obvious since
∂t2(e−2t∂2t(16e2tdet At)) = 192e2t− 108et+ 20e−t ≥ 0.
Actually, it would be possible to prove a priori that At is nonnegative. Indeed, in the
Fourier variables, the semigroup of FP is associated with a transport equation for which many elementary explicit computations can be done. In particular, it follows of a formula of Kolmogorov (see [26,1]) that Atadmits an integral representation on which it is obvious
to see that Atis nonnegative.
Then, we consider the following factorization (whose form could be guessed using The-orem 2.2) that can be checked easily using a formal computation software
∀t ≥ 0, Φηt2+i(vξ−vη) = Φ
i(et−1)vξ
1 Φ
(ξ,η)Att(ξ,η)
1 Φ−ivηt .
Since Atis nonnegative for t≥ 0, applying Proposition3.6, we deduce that
∀t ≥ 0, e−t(v∂x−∂2v−∂vv) = ete−(et−1)v∂xe∇·At∇etv∂v.
4.3.2. Exact splitting for the Kramer–Fokker–Planck equation. Now we focus on the Kramer–Fokker–Planck equation
(KFP) ∂tu(t, x, v) + v2u(t, x, v)− ∂v2u(t, x, v) + v∂xu(t, x, v) = 0.
For some discussions about this equation, we refer to [19,2].
Proposition 4.8. The following factorization provides an exact splitting forKFP
∀t ≥ 0, e−t(v2−∂2v+v∂x)= e−12tanh t v2e∇·At∇e− tanh t v∂xe−12tanh t v2 where At is the nonnegative matrix defined by
At= 1 2 αt sinh2t sinh2t sinh 2t , with αt= 1 2(t− (tanh t)(1 − sinh 2t)).
Proof. Applying Proposition 3.6, we just have to check that At is non negative for t≥ 0
and that ∀t ≥ 0, Φvt2+η2+ivξ = Φ 1 2tanh t v2 1 Φ (ξ,η)Att(ξ,η) 1 Φi tanh t vξ1 Φ 1 2tanh t v2 1 .
This factorization, whose form can be deduced a priori using Theorem2.2, can be verified by a formal computation software. Since the second diagonal coefficient of At is
posi-tive, to check that At is non negative, it is enough to prove that det At ≥ 0 for t > 0.
Thus, we conclude this proof observing that by Jensen’s inequality and some elementary trigonometric computations det At= 1 4tanh(t)(t− tanh(t)) ≥ 0. 5. Appendix
5.1. Proof of Lemma 3.7. Here, the proof relies essentially on computations by block requiring to introduce a more convenient basis than the canonical basis of C2n+2 denoted by e1, . . . , e2n+2. This basis, denoted B, is just a permutation of the canonical basis and
is defined by
(37) B= (e1, . . . , en, en+2, . . . , e2n, e2n+1, en+1, e2n+2). In this basis the matrix of J2n+2 is
matB J2n+2 =
J2n
J2
and the matrix of Ppj is
matB Ppj = Q (j) 1 2 tL(j) 1 2L(j) cj 0 , Consequently, in this basis the matrix of the Hamiltonian map is
matB J2n+2Ppj = J2nQ(j) 12ΥqjJ2n tL(j) 0 −1 2L(j) −cj 0 .
Here, it is really relevant to observe the double triangular structure of this matrix. The four block on the top left corner defines an upper triangular matrix by blocks, whereas, considering these four blocks as a single block, the global matrix is lower triangular by blocks.
Now observe, through the power expansion series, that if Ψ is an entire function and M =
A B
0
is an upper triangular matrix by blocks then Ψ(M ) = Ψ(A) Ψ(z)−Ψ(0) z (A)B Ψ(0) ! . Consequently, we have matB ΦPp1 j = Φ qj 1 −iΥqjJ2n t L(j) 1 iL(j)Υqj 2i eκj+ 2icj 1 where e κj =− i 2L (j)Θ qjJ2n t L(j) with Θqj = ez− 1 − z z2 (−2iJ2nQ(j)).
At the end of the proof, we will check that eκj = κj, so for the moment assume that this
relation holds.
Thus, realizing a product by block we get by a straightforward induction
matB m Y j=1 ΦPpj 1 = m Y j=1 Φqj 1 −i m X j=1 Y k<j Φqk 1 ΥqjJ2n tL(j) 1 i m X j=1 L(j)Υqj Y k>j Φqk 1 2i m X j=1 κj+ cj+ σj 1 .
Identifying the blocks (1, 1), (3, 1), (3, 2) with those of matB ΦPp1 m+1 we get the system
(24). Conversely, we have to check that if (24) is satisfied then the blocks (1, 2) are the same. Indeed, consider a complex symplectic matrix M ∈ Sp2n(C) with a block structure of the form matB M = A B 1 C d 1 .
Note that since M is symplectic, M is invertible and consequently A is also invertible. Since M is symplectic, then is satisfies
However, due to the double triangular nature of M , its invert can be computed easily and we have matB M−1= A −1 −A−1B 1 −CA−1 CA−1B− d 1 . Consequently, a straightforward block product leads to
−matBJ2ntM−1J2n = −J2n tA−1J 2n −J2ntA−1 tC 1 −tBtA−1J 2n d− CA−1B 1 . Thus, since M is symplectic, we have
B =−J2ntA−1 tC.
A fortiori, if two symplectic matrices have this block structure and the same top left corner blocks, if their blocks (3, 1) are equal then their blocks (1, 2) are equal. Consequently, applying this to the symplectic matrices ΦPp1
1 . . . ΦPp1 m and ΦPp1 m+1, we deduce that if the
system (24) is satisfied then we have the factorization (23).
Finally, we just have to check that eκj = κj. For this computation, we omit the indices
j since they are clearly irrelevant. First, we split the even indices from the odd indices in the power expansion of eκ :
2ieκ = LΘqJ2ntL = X k∈N 1 (k + 2)!L(−2iJ2nQ) kJ 2ntL =X k∈N 1 (2k + 2)!L(−2iJ2nQ) 2kJ 2ntL + X k∈N 1 (2k + 3)!L(−2iJ2nQ) 2k+1J 2ntL := Σeven+ Σodd. Observing that (J2nQ)2kJ2n = (J2nQ)kJ2n(QJ2n)k,
since J2n is skew-symmetric we have
L(J2nQ)2kJ2ntL = (−1)kL(J2nQ)kJ2n t
L(J2nQ)k= 0.
Consequently, Σeven vanishes. Similarly, since we have
(J2nQ)2k+1J2n= (J2nQ)kJ2nQ(J2nQ)kJ2n= (−1)k+1(J2nQ)kJ2nQ t ((J2nQ)kJ2n), we get Σodd= 2i X k∈N 4k (2k + 1)!L(J2nQ) kJ 2nQt((J2nQ)kJ2n)tL = 2iκ.
5.2. Proof of Theorem 1.3. Before proving Theorem 1.3, let us to prove some prepara-tory lemmas.
Lemma 5.1. If L is a bounded operator on L2(Rn) such that there exists T ∈ Sp 2n(R)
